Let $\mathcal{X} \in \mathbb{R}^N$ be a random variable with probability density function $p$.
Further assume that we partition $\mathbb{R}^N$ into an infinite number of sets $\mathbb{R}^N = \dot{\bigcup}_i A_i$, each of which is described by a function $a_i : \Theta_i \rightarrow \mathbb{R}^N$, where $\Theta_i \subseteq \mathbb{R}^{D_i}$ is some
$D_i$-dimensional parameter space (think curve or surface, but potentially higher dimensional). That is, $A_i = \left\{a_i(\theta) \mid \theta \in \Theta_i\right\}$.
Note that the $A_i$ may have measure zero.
Define the random variable $\mathcal{Z}$ with $\mathcal{Z} = A_i \iff \mathcal{X} \in A_i$ that returns the subset that random variable $\mathcal{X}$ is in.
My question: How does one determine the distribution / probability density function of $Z$?
If it is not always possible: Under which conditions is it possible to determine the distribution / probability density function of $Z$.
Example: We partition $\mathbb{R}^2$ into concentric circles of different radii: $\mathbb{R}^2 = \dot{\bigcup}_{r \in \mathbb{R}_+} C_r$ with $C_r = \left\{ \begin{bmatrix} r \cos(\theta) & r \sin(\theta) \end{bmatrix}^T \mid \theta \in [0,2\pi] \right\}.$
We can now find the density function $q$ of the distribution over circles $C_r$ by performing a change of variables into polar coordinates and marginalizing out $\theta$, i.e. $$q(r,\theta) = r p\left( \begin{bmatrix} r \cos(\theta) & r \sin(\theta) \end{bmatrix}^T \right) $$ $$q(r) = \int_{0}^{2\pi} r p\left( \begin{bmatrix} r \cos(\theta) & r \sin(\theta) \end{bmatrix}^T \right) \ d \theta. $$
Alternatively, we can compute the line integral over $C_r$, which leads to the same result (but I am not sure if this is just coincidence): $$ \int_{C_r} p(a) \ ds = \int_{0}^{2\pi} p\left( \begin{bmatrix} r \cos(\theta) & \sin(\theta) \end{bmatrix}^T \right) \left\lvert\left\lvert \begin{bmatrix} -r \sin(\theta) \\ r \cos(\theta) \end{bmatrix} \right\rvert\right\rvert_2 \ d\theta $$
I am wondering if either of the two approaches can be generalized for more complicated partitionings.